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Theorem imageval 24469
Description: The image functor in maps-to notation. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
imageval  |- Image R  =  ( x  e.  _V  |->  ( R " x ) )
Distinct variable group:    x, R

Proof of Theorem imageval
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funimage 24467 . . 3  |-  Fun Image R
2 funrel 5272 . . 3  |-  ( Fun Image R  ->  Rel Image R )
31, 2ax-mp 8 . 2  |-  Rel Image R
4 mptrel 24124 . 2  |-  Rel  (
x  e.  _V  |->  ( R " x ) )
5 vex 2791 . . . . 5  |-  y  e. 
_V
6 vex 2791 . . . . 5  |-  z  e. 
_V
75, 6breldm 4883 . . . 4  |-  ( yImage
R z  ->  y  e.  dom Image R )
8 fnimage 24468 . . . . 5  |- Image R  Fn  { x  |  ( R
" x )  e. 
_V }
9 fndm 5343 . . . . 5  |-  (Image R  Fn  { x  |  ( R " x )  e.  _V }  ->  dom Image R  =  { x  |  ( R "
x )  e.  _V } )
108, 9ax-mp 8 . . . 4  |-  dom Image R  =  { x  |  ( R " x )  e.  _V }
117, 10syl6eleq 2373 . . 3  |-  ( yImage
R z  ->  y  e.  { x  |  ( R " x )  e.  _V } )
125, 6breldm 4883 . . . 4  |-  ( y ( x  e.  _V  |->  ( R " x ) ) z  ->  y  e.  dom  ( x  e. 
_V  |->  ( R "
x ) ) )
13 eqid 2283 . . . . . 6  |-  ( x  e.  _V  |->  ( R
" x ) )  =  ( x  e. 
_V  |->  ( R "
x ) )
1413dmmpt 5168 . . . . 5  |-  dom  (
x  e.  _V  |->  ( R " x ) )  =  { x  e.  _V  |  ( R
" x )  e. 
_V }
15 rabab 2805 . . . . 5  |-  { x  e.  _V  |  ( R
" x )  e. 
_V }  =  {
x  |  ( R
" x )  e. 
_V }
1614, 15eqtri 2303 . . . 4  |-  dom  (
x  e.  _V  |->  ( R " x ) )  =  { x  |  ( R "
x )  e.  _V }
1712, 16syl6eleq 2373 . . 3  |-  ( y ( x  e.  _V  |->  ( R " x ) ) z  ->  y  e.  { x  |  ( R " x )  e.  _V } )
18 imaeq2 5008 . . . . . 6  |-  ( x  =  y  ->  ( R " x )  =  ( R " y
) )
1918eleq1d 2349 . . . . 5  |-  ( x  =  y  ->  (
( R " x
)  e.  _V  <->  ( R " y )  e.  _V ) )
205, 19elab 2914 . . . 4  |-  ( y  e.  { x  |  ( R " x
)  e.  _V }  <->  ( R " y )  e.  _V )
215, 6brimage 24465 . . . . 5  |-  ( yImage
R z  <->  z  =  ( R " y ) )
22 eqcom 2285 . . . . . 6  |-  ( z  =  ( R "
y )  <->  ( R " y )  =  z )
2318, 13fvmptg 5600 . . . . . . . . 9  |-  ( ( y  e.  _V  /\  ( R " y )  e.  _V )  -> 
( ( x  e. 
_V  |->  ( R "
x ) ) `  y )  =  ( R " y ) )
245, 23mpan 651 . . . . . . . 8  |-  ( ( R " y )  e.  _V  ->  (
( x  e.  _V  |->  ( R " x ) ) `  y )  =  ( R "
y ) )
2524eqeq1d 2291 . . . . . . 7  |-  ( ( R " y )  e.  _V  ->  (
( ( x  e. 
_V  |->  ( R "
x ) ) `  y )  =  z  <-> 
( R " y
)  =  z ) )
26 funmpt 5290 . . . . . . . . 9  |-  Fun  (
x  e.  _V  |->  ( R " x ) )
27 df-fn 5258 . . . . . . . . 9  |-  ( ( x  e.  _V  |->  ( R " x ) )  Fn  { x  |  ( R "
x )  e.  _V } 
<->  ( Fun  ( x  e.  _V  |->  ( R
" x ) )  /\  dom  ( x  e.  _V  |->  ( R
" x ) )  =  { x  |  ( R " x
)  e.  _V }
) )
2826, 16, 27mpbir2an 886 . . . . . . . 8  |-  ( x  e.  _V  |->  ( R
" x ) )  Fn  { x  |  ( R " x
)  e.  _V }
2920biimpri 197 . . . . . . . 8  |-  ( ( R " y )  e.  _V  ->  y  e.  { x  |  ( R " x )  e.  _V } )
30 fnbrfvb 5563 . . . . . . . 8  |-  ( ( ( x  e.  _V  |->  ( R " x ) )  Fn  { x  |  ( R "
x )  e.  _V }  /\  y  e.  {
x  |  ( R
" x )  e. 
_V } )  -> 
( ( ( x  e.  _V  |->  ( R
" x ) ) `
 y )  =  z  <->  y ( x  e.  _V  |->  ( R
" x ) ) z ) )
3128, 29, 30sylancr 644 . . . . . . 7  |-  ( ( R " y )  e.  _V  ->  (
( ( x  e. 
_V  |->  ( R "
x ) ) `  y )  =  z  <-> 
y ( x  e. 
_V  |->  ( R "
x ) ) z ) )
3225, 31bitr3d 246 . . . . . 6  |-  ( ( R " y )  e.  _V  ->  (
( R " y
)  =  z  <->  y (
x  e.  _V  |->  ( R " x ) ) z ) )
3322, 32syl5bb 248 . . . . 5  |-  ( ( R " y )  e.  _V  ->  (
z  =  ( R
" y )  <->  y (
x  e.  _V  |->  ( R " x ) ) z ) )
3421, 33syl5bb 248 . . . 4  |-  ( ( R " y )  e.  _V  ->  (
yImage R z  <->  y (
x  e.  _V  |->  ( R " x ) ) z ) )
3520, 34sylbi 187 . . 3  |-  ( y  e.  { x  |  ( R " x
)  e.  _V }  ->  ( yImage R z  <-> 
y ( x  e. 
_V  |->  ( R "
x ) ) z ) )
3611, 17, 35pm5.21nii 342 . 2  |-  ( yImage
R z  <->  y (
x  e.  _V  |->  ( R " x ) ) z )
373, 4, 36eqbrriv 4782 1  |- Image R  =  ( x  e.  _V  |->  ( R " x ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   {cab 2269   {crab 2547   _Vcvv 2788   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   "cima 4692   Rel wrel 4694   Fun wfun 5249    Fn wfn 5250   ` cfv 5255  Imagecimage 24383
This theorem is referenced by:  fvimage  24470
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-eprel 4305  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-symdif 24362  df-txp 24395  df-image 24405
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